留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

胞映射方法的研究和进展

徐伟 孙春艳 孙建桥 贺群

徐伟, 孙春艳, 孙建桥, 贺群. 胞映射方法的研究和进展[J]. 力学进展, 2013, 43(1): 91-100. doi: 10.6052/1000-0992-12-022
引用本文: 徐伟, 孙春艳, 孙建桥, 贺群. 胞映射方法的研究和进展[J]. 力学进展, 2013, 43(1): 91-100. doi: 10.6052/1000-0992-12-022
XU Wei, SUN Chunyan, SUN Jianqiao, HE Qun. DEVELOPMENT AND STUDY ON CELL MAPPING METHODS[J]. Advances in Mechanics, 2013, 43(1): 91-100. doi: 10.6052/1000-0992-12-022
Citation: XU Wei, SUN Chunyan, SUN Jianqiao, HE Qun. DEVELOPMENT AND STUDY ON CELL MAPPING METHODS[J]. Advances in Mechanics, 2013, 43(1): 91-100. doi: 10.6052/1000-0992-12-022

胞映射方法的研究和进展

doi: 10.6052/1000-0992-12-022
基金项目: 国家自然科学基金资助项目(11172233,10932009)
详细信息
    作者简介:

    徐伟, 博士, 西北工业大学教授, 专业方向: 非线性动力学, 专长: 随机动力学.

    通讯作者:

    徐伟

  • 中图分类号: O175.1

DEVELOPMENT AND STUDY ON CELL MAPPING METHODS

Funds: The project was supported by the National Natural Science Foundation of China (11172233, 10932009).
More Information
    Corresponding author: XU Wei
  • 摘要: 介绍了胞映射方法的研究和进展. 归纳了目前胞映射方法的几种主要研究方法, 主要包括简单胞映射、广义胞映射、图胞映射、图胞映射的符号分析方法、图胞映射的面向集合方法、邻接胞映射、庞加莱型的简单胞映射、插值胞映射以及胞参照点映射方法, 分析了各类方法的基本特点和特色, 简述了这几种胞映射方法的最新国内外进展, 综述了胞映射方法在控制及相关领域的应用研究及进展, 给出了胞映射方法研究的一些展望, 提出了胞映射方法研究可能率先突破的几个研究方向.

     

  • 1 Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983
    2 Ott E. Chaos in Dynamical Systems. Cambridge: Cambridge University Press, 1993
    3 Moon F C. Chaotic and Fractal Dynamics. New York: John Wiley & Sons, 1992
    4 Wiggins S. Global Bifurcations and Chaos: Analytical Methods. Beijing: Springer-Verlag, 1990
    5 Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc., 1957, 73: 747-753
    6 张芷芬. 微分方程定性理论. 北京: 科学出版社, 1985
    7 Parker T S, Chua L O. Practical Numerical Algorithms for Chaotic Systems. New York: Springer-Verlag, 1989
    8 Hsu C S. A theory of cell-to-cell mapping dynamical systems. J. Applied Mechanics, 1980, 147: 931-939
    9 Hsu C S. Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. New York: Springer-Verlag,1987
    10 Hsu C S. Global analysis by cell mapping. Int. J. Bifurcation and Chaos, 1992, 2(4): 727-771  
    11 Hsu C S. Global analysis of dynamical systems using posets and digraphs. Int. J. Bifurcation and Chaos, 1995,5: 1085-1118  
    12 Hong L, Xu J X. Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys. Lett. A, 1999, 262: 361-375  
    13 Hong L, Xu J X. Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method. Int. J. Bifurcation and Chaos,2001, 11: 723-736  
    14 Tongue B H, Gu K Q. Interpolated cell mapping of dynamical systems. J. Applied Mechanics, 1988, 55: 461-466  
    15 Golat M, Flashner H. A new methodology for the analysis of periodic systems. Nonlinear Dynamics, 2002, 28: 29-51  
    16 Jiang J, Xu J X. A method of point mapping under cell reference for global analysis of nonlinear dynamical systems. Phys. Lett. A, 1994, 188: 137-145  
    17 凌复华. 非线性动力学系统的数值研究. 上海: 上海交通大 学出版社, 1989
    18 Hsu C S. A generalized theory of cell-to-cell mapping for nonlinear dynamical systems. J. Applied Mechanics, 1981,48: 634-642  
    19 Bestle D, Kreuzer E. Modification and extension of an algorithm for generalized cell mapping. Computer Methods in Applied Mechanics and Engineering, 1986, 59: 1-9  
    20 Levitas J. Global stability analysis of fuzzy controllers using cell mapping methods. Fuzzy Sets and Systems, 1999,106: 85-97  
    21 徐健学, 洪灵. 全局分析的广义胞映射图论方法. 力学学报,1999, 31(6): 724-730
    22 洪灵, 徐健学. 两参量平面上双重激变尖点研究. 物理学报,2002, 51(12): 2694-2701
    23 Hong L, Sun J Q. Bifurcations of fuzzy nonlinear dynamical systems. Communications in Nonlinear Science and Numerical Simulation, 2006, 11: 1-12  
    24 Hong L, Sun J Q. Bifurcations of forced oscillators with fuzzy uncertainties by the generalized cell mapping method. Chaos Solitions & Fractals, 2006, 27: 895-904  
    25 Hong L, Sun J Q. Codimension two bifurcations of nonlinear systems driven by fuzzy noise. Physica D, 2006, 213:181-189  
    26 Osipenko G. Dynamical Systems, Graphs, and Algorithms. Berlin: Springer-Verlag 2007
    27 Osipenko G, Ayter S, Kobyakov S. The structure matrix of dynamical system: Tools for mathematical modeling. Mathematical Research, 2001, 8: 06-114
    28 Osipenko G, Pehlivan S. Verification of structural stability, tools for mathematical modeling. Mathematical Research,2001, 8: 115-126
    29 Osipenko G. Calculation of Lyapunov exponents by applied symbolic dynamics. International Journal of Nonlinear Sciences and Numerical Simulation, 2001, 2(1): 53-72
    30 Osipenko G. Spectrum of a dynamical system and applied symbolic dynamics. Journal of Mathematical Analysis and Applications, 2000, 252: 587-616  
    31 Osipenko G, Campbell S. Applied symbolic dynamics: Attractors and filtrations. Discrete and Continuous Dynamical Systems, 1999, 5(1-2): 43-60
    32 Dellnitz M, Hohmann A. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik, 1997, 75: 293-317  
    33 Dellnitz M, Junge O. In: Handbook of Dynamical Systems II: Towards Applications. Singapore: World Scientific,2002. 221-264
    34 Dellnitz M, Froyland G, Junge O. Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Berlin: Springer, 2001. 145-174,
    35 Dellnitz M, Junge O. An adaptive subdivision technique for the approximation of attractors and invariant measures. Comput. Visual. Sci., 1998, 1: 63-68  
    36 Dellnitz M, Junge O. Almost invariant sets in Chua's circuit. Int. J. Bifurcation and Chaos, 1997, 7(11): 2475-2485  
    37 Mehta P G, Hessel-von Molo M, Dellnitz M. Symmetry of attractors and the Perron-Frobenius operator. Journal of Difference Equations and Applications, 2006, 12(11):1147-1178  
    38 Sertl S, Dellnitz M. Global optimization using a dynamical systems approach. Journal of Global Optimization,2006, 34(4): 569-587  
    39 Dellnitz M, Junge O, Koon WS, et al. Transport in dynamical astronomy and multibody problems. Int. J. Bifurcation and Chaos, 2005, 15(3): 699-727  
    40 Dellnitz M, Sch¨utze O, Hestermeyer T. Covering pareto sets by multilevel subdivision techniques. Journal of Optimization, Theory and Applications, 2005, 124(1): 113-136  
    41 Day S, Junge O, Mischaikow K. A Rigorous numerical method for the global analysis of infinite dimensional discrete dynamical systems. SIAM Journal on Applied Dynamical Systems, 2004, 3(2): 117-160  
    42 Junge O, Osinga H. A set oriented approach to global optimal control. ESAIM: Control, Optimisation and Calculus of Variations, 2004, 3(2): 259-270
    43 Froyland G, Dellnitz M. Detecting and locating nearoptimal almost invariant sets and cycles. SIAM Journal on Scientific Computing, 2003, 24(6): 1839-1863  
    44 Dellnitz M, Schutze O, Sertl S. Finding zeros by multilevel subdivision techniques. Journal of Numerical Analysis,2002, 22(2): 167-185  
    45 Dellnitz M, Junge O, Thiere B. The numerical detection of connecting orbits. Discrete and Continuous Dynamical Systems Series B, 2001, 1(1): 125-135  
    46 Froyland G, Junge O, Ochs G. Rigorous computation of topological entropy with respect to a finite partition. Physica D, 2001, 154: 68-84  
    47 Junge O. An adaptive subdivision technique for the approximation of attractors and invariant measures: Proof of convergence. Dynamical Systems, 2001, 16(3): 213-222
    48 贺群, 徐伟, 李爽, 等. 图胞映射的一种改进方法. 物理学报,2008, 57(2): 743-748
    49 贺群, 徐伟, 李爽, 等. 基于复合胞化空间的图胞映射方法. 物理学报, 2008, 57(7): 4021-4028
    50 Yue X L, Xu W. Stochastic bifurcation of an asymmetric single-well potential Duffing oscillator under bounded noise excitation. International Journal of Bifurcation and Chaos, 2010, 20(10): 3359-3371  
    51 Xu W, Yue X L. Global analyses of crisis and stochastic bifurcation in the hardening Helmholtz-Duffing oscillator. Science China Technological Sciences, 2010, 53(3): 664-673  
    52 He Q, Xu W, Rong H W, et al. Stochastic bifurcation in duffing-van der pol oscillators. Physica A, 2004, 338:319-334  
    53 Xu W, He Q, Fang T, et al. Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise. International Journal of Non-Linear Mechanics, 2004, 39: 1473-1479  
    54 Xu W, He Q, Fang T, et al. Global analysis of stochastic bifurcation in Duffing system. International Journal of Bifurcation and Chaos, 2003, 13: 3115-3123  
    55 Zufiria P J, Guttalu R S. The adjoining cell mapping and its recursive unraveling, part one: description of adaptive and recursive algorithms. Nonlinear Dynamics, 1993, 4:207-226
    56 Zufiria P J, Guttalu R S. The adjoining cell mapping and its recursive unraveling, part two: Application to selected problems. Nonlinear Dynamics, 1993, 4: 309-336
    57 Sun J Q, Hsu C S. The generalized cell mapping method in nonlinear random vibration based upon short-time Gaus Gaussian approximation. J. Applied Mechanics, 1990, 57:1018-1025  
    58 Levitas J, Weller T, Singer J. Poincare-like simple cell mapping for nonlinear dynamical systems. J. Sound and Vibration, 1994, 176: 641-662  
    59 Levitas J, Weller T. Poincare linear interpolated cell mapping: method for global analysis of oscillating systems. J. Applied Mechanics, 1995, 62: 489-495  
    60 Hsu C S, Chiu H M. Global analysis of a system with multiple responses including a strange attractor. J. Sound and Vibration, 1987, 114: 203-218  
    61 Zhu W H, Wu Q T. New methods of determining the strange attractor by generalized cell mapping approach. Commun. Appl. Numer. Methods, 1988, 4: 543-548
    62 Jiang J, Xu J X. An iterative method of point mapping under cell reference for the global analysis of nonlinear dynamical systems. J. Sound and Vibration, 1996, 194:605-621  
    63 Jiang J, Xu J X. An iterative method of point mapping under cell reference for the global analysis: Theory and a multiscale reference technique. Nonlinear Dynamics,1998, 15: 103-114  
    64 Guder R, Dellnitz M, Kreuzer E. An adaptive method for the approximation of the generalized cell mapping. Chaos, Solitons and Fractals, 1997, 8(4): 525-534  
    65 Tongue B H, Gu K Q. A higher order method of mapping. J. Sound and Vibration, 1988, 125: 169-179  
    66 文成秀, 姚玉玺, 闻邦椿. 动力系统的点映射胞映射综合 法. 振动工程学报, 1997, 10(4): 413-419
    67 Tongue B H, Gu K Q. A theoretic basis for interpolated cell mapping. SIAM J. Applied Mathematics, 1988, A8:1206-1212
    68 Tongue B H. On obtaining global nonlinear system characteristics through interpolated cell mapping. Physica D,1987, 28: 401-408  
    69 Tongue B H, Gu K Q. A higher order method of interpolated cell mapping. J. Sound and Vibration, 1988, 125:169-179  
    70 Whitf M T, Tongue R H. Application of interpolated cell mapping to analysis of the Lorenz equations. J. Sound and Vibration, 1995, 188(2): 209-226  
    71 Hsu C S. A discrete method of optimal control based upon the cell state space concept. Journal of Optimization Theory and Applications, 1985, 46(4): 547-569  
    72 Bursal F H, Hsu C S. Application of a cell-mapping method to optimal control problems. International Journal of Control, 1989, 49: 1505-1522
    73 Flashner H, Burns T F. Spacecraft momentum unloading: the cell mapping approach. Journal of Guidance, Control and Dynamics, 1990, 13: 89-98  
    74 Zhu W H, Leu M C. Planning optimal robot trajectories by cell mapping. In: Proceedings of Conference on Robotics and Automation, IEEE, New York, 1990. 1730-1735
    75 Wang F Y, Lever P J A. A cell mapping method for general optimum trajectory planning of multiple robotic arms. Robotics and Autonomous Systems, 1994, 12: 15-27  
    76 Yen J Y. Computer disk file track accessing controller design based upon cell to cell mapping. In: Proceedings of the American Control Conference, AACC, 1992
    77 Crespo L G, Sun J Q. Solution of fixed final state optimal control problems via simple cell mapping. Nonlinear Dynamics, 2000, 23: 391-403  
    78 Crespo L G, Sun J Q. Optimal control of target tracking via simple cell mapping. Journal of Guidance, Control and Dynamics, 2000, 24: 1029-1031
    79 Crespo L G, Sun J Q. Fixed final time optimal control via simple cell mapping. Nonlinear Dynamics, 2003, 31:119-131  
    80 Crespo L G, Sun J Q. Stochastic optimal control of nonlinear dynamic systems via bellman's principle and cell mapping. Automatica, 2003, 39: 2109-2114  
    81 Crespo L G, Sun J Q. Stochastic optimal control of nonlinear dynamic systems via short-time gaussian approximation and cell mapping. Nonlinear Dynamics, 2002, 28:323-342  
    82 Crespo L G, Sun J Q. Optimal control of populations of competing species. Nonlinear Dynamics, 2002, 27: 197-210  
    83 Chen Y Y, Tsao T C. Description of the dynamical behavior of fuzzy systems. IEEE Transactions on Systems, Man and Cybernetics, 1989, 19(4): 745-755  
    84 Yen J Y, Chao W C, Lu S S. Fuzzy cell mapping method for a sub-optimal control implementation. Control Engineering Practice, 1994, 2: 247-254.  
    85 Smith S M, Comer D J. Self-tuning of a fuzzy logic controller using a cell state space algorithm. In: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1990.
    86 Smith S M, Corner D J. An algorithm for automated fuzzy logic controller tuning. In: Proceedings of IEEE International Conference on Fuzzy Systems, New York, 1992
    87 Song F, Smith S M. Cell state space based incremental best estimate directed search algorithm for Takagi-Sugeno type fuzzy logic controller automatic optimization. In: Proceedings of the 9th IEEE International Conference on Fuzzy Systems. Taras, 2000
    88 Song F, Smith S M, Rizk C G. Fuzzy logic controller design methodology for 4D systems with optimal global performance using enhanced cell state space based best estimate directed search method. In: Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Tokyo, 1999
    89 Song F, Smith S M, Rizk C G. Optimized fuzzy logic controller design for 4D systems using cell state space technique with reduced mapping error. In: Proceedings of the IEEE International Fuzzy Systems Conference, South Korea, 1999
    90 Edwards D, Choi H T. Use of fuzzy logic to calculate the statistical properties of strange attractors in chaotic systems. Fuzzy Sets and Systems, 1997, 88(2): 205-217  
    91 Sun J Q, Hsu C S. Global analysis of nonlinear dynamical systems with fuzzy uncertainties by the cell mapping method. Computer Methods in Applied Mechanics and Engineering, 1990, 83: 109-120  
    92 Baglio S, Fortuna L, Presti M L. Cube collect: A new strategy to make efficient the classical cell-to-cell algorithm. In: Proceedings of the American Control Conference, New York, 1995
    93 Zufiria P J, Guttalu R S. Adjoining cell mapping and its recursive unraveling, Part I: Description of adaptive and recursive algorithms. Nonlinear Dynamics, 1993, 3: 207-225
    94 Sun J Q. Random Vibrations of Nonlinear Systems Based upon the Cell State Space Concept. [Ph.D. Thesis] Berkeley: University of California, 1988
  • 加载中
计量
  • 文章访问数:  4027
  • HTML全文浏览量:  131
  • PDF下载量:  1832
  • 被引次数: 0
出版历程
  • 收稿日期:  2012-02-28
  • 修回日期:  2012-11-25
  • 刊出日期:  2013-01-24

目录

    /

    返回文章
    返回